Hydrogen Atom Energy: Ratio of νBohr/νorbit

[v_pino]

Homework Statement

In Bohr’s hydrogen atom, the frequency of radiation for transitions between adjacent orbits
is νBohr = (En+1−En)/h. Classically, a charged particle moving in a circular orbit radiates
at the frequency of the motion, νorbit = v/2πr. Find the ratio νBohr/νorbit for the Bohr
orbits as a function of n. Evaluate the ratio for n = 1, 2, 5, 10, 100, and 1000, and thus show that νBohr/νorbit → 1 as n → ∞. This is an example of Bohr’s correspondence principle—that in the limit of large quantum numbers quantum-mechanical results agree with classical results.

Homework Equations

$$\nu_{Bohr}=(E_{n+1}-E_n)/h$$

$$\nu_{orbit}=\frac{v}{2 \pi r}$$

$$E_n= \frac{Z^2}{n^2}E_1$$

The Attempt at a Solution

From equation 3, can I say $E_{n+1}= \frac{Z^2}{(n+1)^2}E_1$ ?

Then doing the algebra gives me

$$\frac{\nu_{Bohr}}{\nu_{Orbit}}=(\frac{1}{(n+1)^2}-\frac{1}{n^2} )\frac{Z^2E_1}{h} \frac{2 \pi r}{v}$$

Does that seem correct?

However, the values from the brackets of $\frac{1}{(n+1)^2}-\frac{1}{n^2}$ gives me -3/4, -5/36, ... which don't tend to 1.

 

[Redbelly98]

Looks right so far. But you have not accounted for the fact that r and v are not constant but depend on n, which will affect the value of the limit for large n. (For example, r is larger for higher n, since higher energy states correspond to larger orbits.)

 

[v_pino]

Now I have $\frac{\nu_{Bohr}}{\nu_{Orbit}}=(\frac{n}{(n+1)^2}-1 )\frac{Z^2E_1}{h} \frac{2 \pi r}{v}$ using $r_n=\frac{n^2}{Z}a_0$.

But even so, this only tells me that the brackets value is getting very small at large n.

 

[Redbelly98]

You need to substitute an expression for v_n as well.

Also, this was probably just a typo on your part, but inside the brackets in your previous post it should be
$$\left( \frac{n^2}{(n+1)^2}-1 \right)$$

 

[v_pino]

I now have the ratio $= (\frac {n^3}{(n+1)^2}-n)\frac{E_1 \hbar^2}{mk^2e^2}$

But when I evaluate the non-bracket side, I get 1.28*10^-38.
Multiplying this to the bracket doesn't give me a value that tends to 1.

 

[Redbelly98]

I now have the ratio $= (\frac {n^3}{(n+1)^2}-n)\frac{E_1 \hbar^2}{mk^2e^2}$

Something is wrong, the units are not working out correctly.

Now I have $\frac{\nu_{Bohr}}{\nu_{Orbit}}=(\frac{n}{(n+1)^2}-1 )\frac{Z^2E_1}{h} \frac{2 \pi r}{v}$ using $r_n=\frac{n^2}{Z}a_0$.

If you have replaced r with n²a₀/Z, then r shouldn't be in your expression, and a₀ should be there. Also, what expression for v_n did you use? If you show more of your steps, we should be able to figure out where things went wrong.

 

[v_pino]

Is $v_n=\frac{Zke^2}{n \hbar}=\frac{Z\alpha c}{n}$?

 

[v_pino]

Is this right? $= (\frac {n^3}{(n+1)^2}-n)\frac{E_1 \hbar^2}{mk^2e^4}$

The non bracket is now very big: 1.045*10^40.

I think the bracket tends to -2.

 

[v_pino]

got it - thanks

 

[Redbelly98]

Is this right? $= (\frac {n^3}{(n+1)^2}-n)\frac{E_1 \hbar^2}{mk^2e^4}$

Looks good.

The non bracket is now very big: 1.045*10^40.

I get something different, perhaps you are not paying attention to units and just plugging in some numbers. So -- try including units in your calculation so that you can see them cancelling properly. If they don't cancel out, you probably have to convert some units somewhere. If it's still not working out after doing that, post your calculation of the non-bracket part here.

I think the bracket tends to -2.

I agree.

 

[Redbelly98]

got it - thanks

Ah, glad it worked out